(1) Background: Contact tracing and notification apps for coronavirus disease 2019 (COVID-19) are installed on smartphones and are intended to detect contact with another person’s device. A high installation rate is important for these apps to enable them to be effective countermeasures against the silent transmission of diseases. However, the installation rate varies among apps and regions and the penetration dynamics of these applications are unclear. (2) Methods: The download behavior of contact tracing applications was investigated using publicly available datasets. The increase in downloads was modeled using a system dynamics model derived from the product growth model. (3) Results: The imitation effects present in the traditional product growth model were not observed in COVID-19 contact tracing apps. The system dynamics model, without the imitation effect, identified the downloads of the Australian COVIDSafe app. The system dynamics model, with a layered adopter, identified the downloads of the Japanese tracing app COCOA. The spread of COVID-19 and overall anti-COVID-19 government intervention measures in response to the spread of infection seemed to result in an increase in downloads. (4) Discussion: The suggested layered structure of users implied that individualized promotion for each layer was important. Addressing the issues among users who are skeptical about adoption is pertinent for optimal penetration of the apps.